Doubly-periodic array of bubbles in a Hele-Shaw cell

TL;DR

This paper derives exact, conformal mapping-based solutions for steady, doubly-periodic bubble arrays in a Hele-Shaw cell, considering symmetry conditions and neglecting surface tension, with applications to various geometries.

Contribution

It provides a general integral-form solution for multiple bubbles per unit cell in Hele-Shaw flows, extending previous work to doubly-periodic configurations with symmetry assumptions.

Findings

Exact solutions for doubly-periodic bubble arrays are obtained.

Solutions include multi-file and unbounded cell configurations.

The approach uses conformal mapping techniques under symmetry constraints.

Abstract

Exact solutions are presented for a doubly-periodic array of steadily moving bubbles in a Hele-Shaw cell when surface tension is neglected. It is assumed that the bubbles either are symmetrical with respect to the channel centreline or have fore-and-aft symmetry, or both, so that the relevant flow domain can be reduced to a simply connected region. By using conformal mapping techniques, a general solution with any number of bubbles per unit cell is obtained in integral form. Several examples are given, including solutions for multi-file arrays of bubbles in the channel geometry and doubly-periodic solutions in an unbounded cell.